A192750 - OEIS

%I #48 Sep 16 2024 12:00:55

%S 1,6,11,21,36,61,101,166,271,441,716,1161,1881,3046,4931,7981,12916,

%T 20901,33821,54726,88551,143281,231836,375121,606961,982086,1589051,

%U 2571141,4160196,6731341,10891541,17622886,28514431,46137321,74651756

%N Define a pair of sequences c_n, d_n by c_0=0, d_0=1 and thereafter c_n = c_{n-1}+d_{n-1}, d_n = c_{n-1}+4*n+2; sequence here is d_n.

%C Old definition was: constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined recursively by p(n,x) = x*p(n-1,x) + 4n+2 for n>0, with p(0,x)=1.

%C For discussions of polynomial reduction, see A192232 and A192744.

%H Harvey P. Dale, <a href="/A192750/b192750.txt">Table of n, a(n) for n = 0..1000</a>

%H Feng-Zhen Zhao, <a href="https://math.colgate.edu/~integers/y82/y82.pdf">The log-behavior of some sequences related to the generalized Leonardo numbers</a>, Integers (2024) Vol. 24, Art. No. A82.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -1).

%F G.f.: ( 1+4*x-x^2 ) / ( (x-1)*(x^2+x-1) ). The first differences are in A022088. - _R. J. Mathar_, May 04 2014

%F a(n) = 5*Fibonacci(n+2)-4. - _Gerry Martens_, Jul 04 2015

%F a(n) = A265752(A265750(n)). - _Antti Karttunen_, Dec 15 2015

%t q = x^2; s = x + 1; z = 40;

%t p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 2;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

%t   (* A192750 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

%t   (* A192751 *)

%t LinearRecurrence[{2,0,-1},{1,6,11},40] (* _Harvey P. Dale_, Dec 03 2023 *)

%Y See A192751 for c_n.

%Y Cf. A000045, A192744, A192232, A022088, A265750, A265752.

%K nonn,changed

%O 0,2

%A _Clark Kimberling_, Jul 09 2011

%E Entry revised by _N. J. A. Sloane_, Dec 15 2015